We add small random perturbations to a cellular automaton and consider the one-parameter family $(F_\epsilon)_{\epsilon>0}$ parameterized by $\epsilon$ where $\epsilon>0$ is the level of noise. The objective of the article is to study the set of limiting invariant distributions as $\epsilon$ tends to zero denoted $\Ml$. Some topological obstructions appear, $\Ml$ is compact and connected, as well as combinatorial obstructions as the set of cellular automata is countable: $\Ml$ is $\Pi_3$-computable in general and $\Pi_2$-computable if it is uniformly approached. Reciprocally, for any set of probability measures $\mathcal{K}$ which is compact, connected and $\Pi_2$-computable, we construct a cellular automaton whose perturbations by an uniform noise admit $\mathcal{K}$ as the zero-noise limits measure and this set is uniformly approached. To finish, we study how the set of limiting invariant measures can depend on a bias in the noise. We construct a cellular automaton which realizes any connected compact set (without computable constraints) if the bias is changed for an arbitrary small value. In some sense this cellular automaton is very unstable with respect to the noise.

翻译：本文通过向元胞自动机引入小随机扰动，研究以噪声强度ε>0为参数的单参数族(F_ε)_{ε>0}。文章的核心目标是分析当ε趋于零时极限不变分布的集合，记为\Ml。研究揭示了若干拓扑约束：\Ml具有紧致性与连通性；同时由于元胞自动机构成的可数性，也存在组合约束：一般情况下\Ml是Π_3可计算的，若满足一致逼近条件则为Π_2可计算。反之，对于任意紧致、连通且Π_2可计算的概率测度集\mathcal{K}，我们都能构造一个元胞自动机，使其在均匀噪声扰动下以\mathcal{K}作为零噪声极限测度集，且该集合可被一致逼近。最后，我们探究极限不变测度集如何随噪声偏置变化。通过构造一个对噪声极度敏感的元胞自动机，证明当噪声偏置发生任意微小改变时，该自动机可实现任意连通紧致集（无需满足可计算性约束）。这从某种意义上表明该元胞自动机对噪声具有高度不稳定性。